An Efficient Computational Method for Large-Scale
Operations Planning in Power Systems
Javad Lavaei
Department of Electrical Engineering
Columbia University
Acknowledgements
Caltech: Steven Low, Somayeh Sojoudi
Columbia University: Ramtin Madani
UC Berkeley: David Tse, Baosen Zhang
Stanford University: Stephen Boyd, Eric Chu, Matt Kranning
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J. Lavaei and S. Low, "Zero Duality Gap in Optimal Power Flow Problem," IEEE Transactions on Power
Systems, 2012.
J. Lavaei, D. Tse and B. Zhang, "Geometry of Power Flows in Tree Networks,“ in IEEE Power & Energy
Society General Meeting, 2012.
S. Sojoudi and J. Lavaei, "Physics of Power Networks Makes Hard Optimization Problems Easy To
Solve,“ in IEEE Power & Energy Society General Meeting, 2012.
M. Kraning, E. Chu, J. Lavaei and S. Boyd, "Message Passing for Dynamic Network Energy
Management," Submitted for publication, 2012.
S. Sojoudi and J. Lavaei, "Semidefinite Relaxation for Nonlinear Optimization over Graphs with
Application to Optimal Power Flow Problem," Working draft, 2012.
S. Sojoudi and J. Lavaei, "Convexification of Generalized Network Flow Problem with Application to
Optimal Power Flow," Working draft, 2012.
Power Networks (CDC 10, Allerton 10, ACC 11, TPS 11, ACC 12, PGM 12)
 Optimizations:
 Resource allocation
 State estimation
 Scheduling
 Issue: Nonlinearities
(power quadratic in voltage)
 Transition from traditional grid to smart grid:
 More variables (10X)
 Time constraints (100X)
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Resource Allocation: Optimal Power Flow (OPF)
Voltage V
Current I
Complex power = VI*=P + Q i
OPF: Given constant-power loads, find optimal P’s subject to:
 Demand constraints
 Constraints on V’s, P’s, and Q’s.
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Broad Interest in Optimal Power Flow
 Interested companies: ISOs, TSOs, RTOs, Utilities, FERC
 OPF solved on different time scales:
 Electricity market
 Real-time operation
 Security assessment
 Transmission planning
 Existing methods based on linearization or local search
 Can save $$$ if solved efficiently
 Huge literature since 1962 by power, OR and Econ people
Recent conference by Federal Energy Regulatory Commission about OPF
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Local Solutions
P1
P2
OPF
Local solution: $1502
Global solution: $338
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Local Solutions
Source of Difficulty: Power is quadratic in terms of complex voltages.
Anya Castillo et al.
Ian Hiskens from Umich:
 Study of local solutions by Edinburgh’s group
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Summary of Results
Project 1: How to solve a given OPF in polynomial time? (joint work with Steven Low)
 A sufficient condition to globally solve OPF:
 Numerous randomly generated systems
 IEEE systems with 14, 30, 57, 118, 300 buses
 European grid
 Various theories: It holds widely in practice
Project 2: Find network topologies over which optimization is easy? (joint work with Somayeh
Sojoudi, David Tse and Baosen Zhang)
 Distribution networks are fine.
 Every transmission network can be turned into a good one.
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Summary of Results
Project 3: How to design a distributed algorithm for solving OPF? (joint work with Stephen Boyd,
Eric Chu and Matt Kranning)
 A practical (infinitely) parallelizable algorithm
 It solves 10,000-bus OPF in 0.85 seconds on a single core machine.
Project 4: How to do optimization for mesh networks? (joint work with Ramtin Madani)
Project 5: How to relate the polynomial-time solvability of an optimization to its
structural properties? (joint work with Somayeh Sojoudi)
Project 6: How to solve generalized network flow (CS problem)? (joint work with Somayeh
Sojoudi)
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Geometric Intuition: Two-Generator Network
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Optimal Power Flow
Cost
Operation
Flow
Balance
 Extensions:
 Other objective (voltage support, reactive power, deviation)
 More variables, e.g. capacitor banks, transformers
 Preventive or corrective contingency constraints
 Multi-period OPF
 Conventional OPF captures common sources of non-convexity.
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Optimal Power Flow
 Express balance equations as inequalities:
 Proof without this change involves geometric techniques.
 Allow power over-delivery or assume positive LMPs.
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Various Relaxations
OPF
Dual OPF
SDP
R-SDP
SOCP
 SDP relaxation has a rank-one solution for:
 IEEE systems with 14, 30, 57, 118, 300 buses
 European grid
 Exactness of SDP relaxation and zero duality gap are equivalent for OPF.
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Zero Duality Gap
 OPF:
 Real-valued (DC)
 Complex-valued (AC)
 Networks:
 Distribution (acyclic)
 Transmission (cyclic)
 DC lines are becoming more deployed (Nordic circuit).
 Theory applies to scheduling of EVs charging, control of PV invertors,...
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AC Transmission Networks
 How about AC transmission networks?
 May not be true for every network
 Various sufficient conditions
 Result 1: AC transmission network manipulation:
 High performance (lower generation cost)
 Easy optimization
 Easy market (positive LMPs and existence of eq. pt.)
Bus-4
PS
Bus-2
Bus-3
Bus-5
 Result 2: Reduced computational complexity
Bus-1
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Simulations
Simulations:
 Zero duality gap for IEEE 30-bus system
 Guarantee zero duality gap for all possible load profiles?
 Theoretical side: Add 12 phase shifters
 Practical side: 2 phase shifters are enough
 IEEE 118-bus system needs no phase shifters (power loss case)
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OPF With Equality Constraints
 Injection region under fixed voltage magnitudes:
 When can we allow equality constraints? Need to study Pareto front
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Convexification in Polar Coordinates
Similar to the condition derived in Ross Baldick’s book
 Imposed implicitly (thermal, stability, etc.)
 Imposed explicitly in the algorithm
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Convexification in Polar Coordinates
 Idea:
 Algorithm:
 Fix magnitudes and optimize phases
 Fix phases and optimize magnitudes
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Convexification in Polar Coordinates
 Can we jointly optimize phases and magnitudes?
Change of variables:
Assumption (implicit or explicit):
 Observation 1: Bounding |Vi| is the same as bounding Xi.
 Observation 2:
is convex for a large enough m.
 Observation 3:
is convex for a large enough m.
 Observation 4: |Vi|2 is convex for m ≤ 2.
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Lossy Networks
 Relationship between polar and rectangular?
 Assumption (implicit or explicit):
 Conjecture: This assumptions leads to convexification in rectangular coordinates
(partially proved).
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Lossless Networks
 Consider a lossless AC transmission network.
Lossless 3 bus
(P12,P23,P31)
(P1,P2)
 SDP does not work well on lossless networks.
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Lossless Networks
Solution: Penalize the rank constraint in the objective:
 Simulation for a 10-bus cycle:
 1) W has 8 nonzero eigs, but the regularization kills 7 of all (zero duality gap).
 2) This regularization is guaranteed to kill all but 1-2 eigs.
 3) Randomized “M” gives us the correct answer (nonzero duality gap).
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Conclusions

Focus: OPF with a 50-year history

Goal: Find a global solution efficiently
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Obtained provably global solutions for many practical OPFs
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Developed various theories for distribution and transmission networks
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Several companies working on these ideas
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